If I were to speculate, I suspect the differences are:
1. Modestly higher pay in the private school, attracting better teachers
2. More selective student body in the private school, making the school environment more conducive to learning for average and above-average students, and more attractive to teachers
3. No public-sector union in the private school, leading to more accountability for teachers
The second point is important. Private schools exclude all the kids whose parents lack the motivation, time, or resources to place their child in an exclusive institution. They exclude the children who didn't learn the behavioral skills to conform to the expectations of a private school. They exclude many children with disabilities even if they can't explicitly discriminate. Meanwhile, public schools are required to include children with disabilities (often involving disruptive behavior) in class with other students to the extent practical.
Public schools are charged with upholding the social contract that all children are entitled to a Free Appropriate Public Education regardless of their social status, wealth, disability, or the financial or emotional capacity of their parents. It's a heavy burden to carry, but it's one that we as a society have decided is worth the cost. Your children's private school doesn't carry that burden.